Here, we consider a particular measure of fairness which has gained attention in the research community [1]}, [2]}, the \(\alpha \) -fairness scheme, that was introduced by [3]}. According to this scheme, the decision maker decides on the allocation of the resources by maximizing the constant elasticity social welfare operator \(\Psi _\alpha \)  parameterized by \(\alpha \ge 0\)  and for \(\mathbf {W} = (W_1,\ldots ,W_p) \in \mathbb {R}^p_{+}\) , the amount of resources allocated into \(p\)  agents,
\(\Psi _\alpha (W_1, \ldots , W_p) = \left\lbrace \begin{array}{cl}\frac{1}{1-\alpha } \displaystyle \sum _{j =1}^p W_j^{1-\alpha } & \mbox{if $\alpha \ge 0$, $\alpha \ne 1$,}\\\displaystyle \sum _{j=1}^p \log (W_j) & \mbox{if $\alpha =1$}.\end{array}\right.\) 
